In Compound Interest, compounding more often gave a little more money. What if a bank compounded every month, every day, every second, without limit? The answer leads to one of the most important numbers in mathematics.
Key idea
As $n$ grows, $\left(1 + \frac{1}{n}\right)^n$ approaches $e \approx 2.71828$. Compounding continuously at rate $r$ for $t$ years gives $$A = Pe^{rt}.$$ The function $e^x$ is the natural exponential function.
Why it works
With rate $r$ compounded $n$ times a year, $\left(1 + \frac{r}{n}\right)^{nt}$. Write $m = \frac{n}{r}$: this becomes $\left[\left(1 + \frac{1}{m}\right)^{m}\right]^{rt}$. As $n$ grows, $m$ grows, and the inner bracket approaches $e$. So the limit of ever-more-frequent compounding is $e^{rt}$. The values settle down because each extra split of the year adds less and less.
| $n$ | 1 | 2 | 12 | 365 | 1,000,000 |
|---|---|---|---|---|---|
| value | 2 | 2.25 | 2.6130 | 2.7146 | 2.7183 |
Worked examples
Worked example: Continuous compounding
\$500 grows at 4% per year, compounded continuously, for 10 years. Find the balance.
- $A = 500e^{0.04 \cdot 10} = 500e^{0.4}$.
- $e^{0.4} \approx 1.49182$, so $A \approx 745.91$.
Answer: about \$745.91
Worked example: Continuous versus monthly
Compare with monthly compounding for the same account.
- Monthly: $500\left(1 + \frac{0.04}{12}\right)^{120} \approx 745.42$.
Answer: Continuous gives about \$0.50 more: the most any compounding could give.
$e$ shows up far beyond banking. The function $e^x$ is the one whose rate of growth at every moment equals its current value, which is why it describes continuous growth and decay, and why calculus uses it constantly. Like $\pi$, $e$ is irrational: its decimals never end or repeat.
Common mistake
Typing $e^{0.4}$ as $e \cdot 0.4$ or using $e \approx 3$. Use your calculator's $e^x$ key, and keep the rate as a decimal ($4\% = 0.04$). A quick estimate: $e^{0.4}$ should be between $1.4$ and $1.5$.